Two Compact Incremental Prime Sieves

Abstract

A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper we present two new results: (1) We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log \log n)$ time and $O(\sqrt{n}\log n)$ bits of space, and (2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieve that is simultaneously sublinear, compact, and incremental. The second result solves an open problem given by Paul Pritchard in 1994.

References in corpus (1)

• Approximately counting semismooth integers

Cited by in corpus (2)

• Computation of the least primitive root
• Reducing the Space Used by the Sieve of Eratosthenes When Factoring